Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > sbalv | Structured version Visualization version GIF version |
Description: Quantify with new variable inside substitution. (Contributed by NM, 18-Aug-1993.) |
Ref | Expression |
---|---|
sbalv.1 | ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) |
Ref | Expression |
---|---|
sbalv | ⊢ ([𝑦 / 𝑥]∀𝑧𝜑 ↔ ∀𝑧𝜓) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sbal 2156 | . 2 ⊢ ([𝑦 / 𝑥]∀𝑧𝜑 ↔ ∀𝑧[𝑦 / 𝑥]𝜑) | |
2 | sbalv.1 | . . 3 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜓) | |
3 | 2 | albii 1811 | . 2 ⊢ (∀𝑧[𝑦 / 𝑥]𝜑 ↔ ∀𝑧𝜓) |
4 | 1, 3 | bitri 276 | 1 ⊢ ([𝑦 / 𝑥]∀𝑧𝜑 ↔ ∀𝑧𝜓) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 ∀wal 1526 [wsb 2060 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-11 2151 |
This theorem depends on definitions: df-bi 208 df-ex 1772 df-sb 2061 |
This theorem is referenced by: sbex 2279 sbmo 2691 sbabel 3012 mo5f 30180 |
Copyright terms: Public domain | W3C validator |